Some s-numbers of embeddings of function spaces with weights of logarithmic type

In this paper we deal with compact embeddings of weighted function spaces of Besov and Triebel-Lizorkin type with weights of logarithmic growth near infinity. We obtain the exact estimates for the asymptotic behaviour of Gelfand, Kolmogorov and Weyl numbers of the embeddings. As an application we get eigenvalue estimates of some degenerate pseudodifferential operators of type. Finally we deal with the negative spectrum of operators H-gamma = A - gamma V for gamma -> infinity, where the potential V is a positive function and A is a positive elliptic pseudodifferential operator. We concentrate on the case when the Weyl numbers and Pietsch's inequality give us better estimates then entropy numbers and Carl's inequality. (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim